This study will examine the intrinsic difficulty of estimating subpopulation characteristics when the identities of the observations are not completely known. Hard clustering and fuzzy clustering techniques to classify observations into subgroups are investigated. This study will develop estimation methodologies for the subpopulation characteristics. The statistical properties of the new methodologies will be studied. The new methodologies will be compared with the existing methodologies designed to remedy the misclassification problem. The new methodologies will alleviate some of the intrinsic difficulties for a heterogeneous mixture of subpopulations. Applications will be made to study the issues concerning biological heterogeneity and measurement error in the birth weight, gestational age and perinatal mortality. This study will also be applied to the investigation of intrauterine growth and pregnancy outcome. Another topic in mixture of distributions is to develop and compare parametric and nonparametric tests to assess homogeneity assumption in mixed models for clinical trials and other biomedical studies. Estimation of mixing parameters will be investigated. The methods developed in Yu (1991, comm Stat) will be extended to a mixing distribution of K subpopulation from a mixture of two subpopulations. Numerical algorithms will be studied.